Trying to add uv project modifier on eye but eye gets stretched up and down
The reason the texture isn't projecting right may be due to the lack of Coordinate and Mapping nodes, like so:

With the Texture Coordinate set to UV, the UV Project and UV Warp modifiers should start functioning properly. You can quickly add these nodes in with the built-in Node Wrangler add-on, using Ctrl+T after selecting the image texture node.
Note that in this example, I have the X and Y scale on the Mapping node set to -1, in order to flip the centered texture along those axis. Those properties can be animated as well, but the eye rig should be able to take over from there.